Rate processes Kramers

The Langevin equation, in which the time-dependent friction is ohmic, plays a special role in the theory of activated rate processes. Kramers originally formulated the problem in terms of ohmic friction. In most applications in chemistry the friction will not be ohmic however, the ohmic case is the simplest to analyze. Apart from the historic importance, the analytical simplicity helps in understanding and analyzing more difficult cases of space- and time-dependent friction. A summary of important results for the parabolic barrier and ohmic friction is presented in Sec. III.C. 

Berezhkovskii A M and Zitserman V Yu 1991 Activated rate processes in the multidimensional case. Consideration of recrossings in the multidimensional Kramers problem with anisotropic friction Chem. Phys. 157 141-55... 

Nitzan A 1988 Activated rate processes in condensed phases the Kramers theory revisited Adv. Chem. Phys. 70 489 Onuchic J N and Wolynes P G 1988 Classical and quantum pictures of reaction dynamics in condensed matter resonances, dephasing and all that J. Phys. Chem. 92 6495... 

See, for example, Poliak E 1986 Theory of activated rate processes a new derivation of Kramers expression J. Chem. Phys. 85 865... 

Berezhkovskii A M, Poliak E and Zitserman V Y 1992 Activated rate processes generalization of the Kramers-Grote-Hynes and Langer theories J. Chem. Phys. 97 2422... 

The standard language used to describe rate phenomena in condensed phases has evolved from Kramers one dimensional model of a particle moving on a one dimensional potential, feeling a random and a related friction force. In Section II, we will review the classical Generalized Langevin Equation (GEE) underlying Kramers model and its application to condensed phase systems. The GLE has an equivalent Hamiltonian representation in terms of a particle which is bilinearly coupled to a harmonic bath. The Hamiltonian representation, also reviewed in Section II is the basis for a quantum representation of rate processes in condensed phases. Eas also been very useful in obtaining solutions to the classical GLE. Variational estimates for the classical reaction rate are described in Section III. 

For example, Ber. Bunsenges. Phys. Chem. 95, no.3 (1991), Rate Processes in Dissipative Systems. H.A. Kramers, Physica 7, 284 (1940),... 

ACTIVATED RATE PROCESSES IN CONDENSED PHASES THE KRAMERS THEORY REVISITED... 

This chapter reviews the generalizations of the Kramers model that were develojjed during the past few years. The result of this effort, which we may call the generalized Kramers theory, provides a useful framework for the theoretical description of activated rate processes in general and of chemical reaction rates in condensed phases in particular. Some applications of this framework as well as its limitations are also discussed. In the last few years there has also been substantial progress in the study of the quantum mechanical Kramers model, which may prove useful for condensed phase tunneling reactions. This aspect of the problem is not covered by the present review. 

Although the Kramers model contains much of the essential physics of the activated escajje problem, it cannot be used for quantitative discussion of many realistic activated processes. In particular the model is too oversimplified for the original application intended by Kramers for chemical rate processes. The theory needs to be generalized to correct the following shortcomings of the Kramers model. 

The Kramers theory and its extensions have found many applications since the original work by Kramers. Recent application of the non-Markovian theory in the low-friction limit to thermal desorption was described by Nitzan and Carmeli. Another novel application of the Markovian theory is to transition from a nonequilibrium state of a Josephson junction. In what follows we shall briefly review the recent application of the generalized Kramers theory to chemical rate processes. More detailed reviews of the exjjerimental and theoretical status of this field may be found in Hynes. ... 

The starting point of the Kramers theory of activated rate processes is the onedimensional Markovian Langevin equation, Eq. (8.13)... 

Mathematical difficulties forced Kramers to restrict his discussion. to the case in which the barrier height Q = EMt is large compared to the mean thermal energy of the molecules kT and in which the diffusion over the barrier can be treated as a quasi-stationary process. Kramers showed that under these conditions the calculated reaction rate is very close to the equilibrium rate, as given by absolute rate theory, and that for E/kT > 10 the rate calculated from his model agrees with the equilibrium rate to within about 10 per cent over a rather wide range of rj. 

Kramers paper spurred an enormous amount of research on the theory of activated rate processes, especially in the physics community, as evidenced in numerous textbooks see, for example, Refs. 13 and 14. However, as noted by Landauer (15) in his subjective description of the history of noise activated escape from metastable states, up till the end of the seventies, the physical chemistry community largely ignored the theory of rates introduced by Kramers. The first experimental measurements of viscosity effects on activated rate processes were performed on the isomerization of frans-stilbene to c/s-stil-bene by Fischer and co-workers in 1968 (16). These authors were not aware of Kramers work and interpreted their results in terms of the free volume necessary for isomerization to occur. Since then, experimental work has proliferated see, for example, the recent textbook (17). 

E. Poliak, J. Chem. Phys., 85, 865 (1986). Theory of Activated Rate Processes A New Derivation of Kramers Expression. 

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